TAGUCHI’S METHODS

Genichi Taguchi

INTRODUCTION

Professor Genichi Taguchi was the director of the Japanese Academy of quality and four times receipt of the Deming Prize.

He starts where SPC (temporarily) finishes. He can help with the identification of common cause of variation, the most difficult to determine and eliminate in process.

He attempts to go even further: he tries to make the process and the product robust against their effect (eliminate of the effect rather then the cause) at the design stage.

Even if the removal of the effect is impossible, he provides a systematic procedure for controlling the noise (through tolerance design) at the minimum cost.

When Dr. Taguchi was first brought his ideas to America in 1980, he was already well known in Japan for his contribution to quality engineering.

Traditional Taguchi’s

There is Good or Bad Products only as per Limits.

When a product moves from its target will cause the loss even if the product lies or not within Limits.

Traditional and Taguchi’s Defination of Quality

Objective of Taguchi Methods

1)Minimize the variation in product response while keeping the mean response on target.

2)The product can be made robust to changes in operating and environmental conditions.

3)Since the method is applied in a systematic way at a pre-production stage (off-line), it can greatly reduce the number of time-consuming tests, thus saving in costs and wasted products.

Example of Robust

Taguchi Loss Function : Definition

“Quality is the loss a product causes to society after being shipped, other then any losses caused by its intrinsic functions.”

By “loss” Taguchi refers to the following two categories:

Loss caused by variability of function. Loss caused by harmful side effects.

An example of loss caused by variability if function would be an automobile that does not start in cold weather. The car’s owner would suffer a loss if he or she had to pay some to start a car. The car owner’s employer losses the services of the employee who is now late for work.

Comparing the Quality Levels of SONY TV Sets Made in JAPAN and in SAN DIEGO

The front page of the Ashi News on April 17,1979 compared the quality levels of Sony color TV sets made in Japanese plants and those made in San Diego, California, plant. The quality characteristic used to compare these sets was the color density distribution, which affect color balance. Although all the color TV sets had the same design, most American customers thought that the color TV sets made in San Diego plant were of lower quality than those made in Japan.

Distribution of the Quality characteristic

Colour density of the TV sets from Japanese Sony plants are normally disturbed around the target value m, and S.D is 10/6.

Color density of San Diego TV sets is uniformly distributed rather than normally distributed. Therefore, the S.D of these uniformly distributed objective characteristics is about 10/√12.

Process Capability study The process capability index(Cp) is defined as the tolerance specification divided by 6 times the standard deviation of the objective characteristic: Cp=Tolerance/6*Standard deviation

The Cp of the of Japanese Sony TV sets is about 1. In addition, the mean value of the distribution of these objective characteristics is very close to the target value of m.

The process Cp of the San Diego Sony plant is calculated as follows:

Cp=Tolerance/6(Tolerance/√12) = 0.577

 It is obvious that the process capability index of San Diego Sony is much lower than that of Japanese Sony.

Tolerance specifications are very similar to tests in schools, where 60% is usually the dividing line between passing and failing, and 100% is ideal score.the grades between 60% and 100% in evaluating quality can be classified as follows:

  60%-69% Passing (D) 70%-79% Fair (C) 80%-89% Good (B) 90%-100% Excellent (A)We can apply this scheme to the classification of the objective characteristics (color density) of these color TV sets as shown in Figure. One can see that a very high percentage of Japanese Sony TV sets are within grade B, and a very low percentage are within or below grade D. In comparison, the color TV sets from San Diego SONY have about the same percentage in grades A, B and C.

To reduce the difference in process capability indices between Japanese SONY and San Diego SONY, (and thus seemingly increase the quality level of the San Diego sets) the letter tried to tighten the tolerance specification to extend only to grade C rather than grade D. Therefore, Only the products within grades A,B and C were treated as passing.

But this approach is faulty, Tightening the tolerance specifications because of a low process capability in a production plant is meaningless as increasing the passing score of school tests from 60% to 70% just because students do not learn well. On the contrary, such a school should consider asking the teachers to lower the passing score for student who do not test as well instead of rating it.

Now we section will discuess how to evaluate the functional quality of products meaningfully and correctly.

Taguchi’s Quadratic Quality Loss Function

Quality Loss Occurs when a product’s deviates from target or nominal value.

Deviation Grows, then Loss increases. Taguchi’s U-shaped loss Function Curve.

Taguchi’s U-shaped loss Function Curve

Measuredcharacteristic

Taguchi loss Fn

LTL Nominal UTL

Scrap or Rework Cost.

Loss

Taguchi’s method: Loss function..

Loss = L(y) = L( m + (y-m))

= L(m) + (y-m) L’(m)/ 1! + (y – m)2 L”(m)/ 2! + …

Taguchi’s Approximation: L(y) ≈ k( y – m)2

Ideally:(a) L(m) = 0 [if actual size = target size, Loss = 0], and

(b) When y = m, the loss is at its minimum, therefore L'(m) = 0

Formula to find Taguchi’s Loss Fn

L (x) = k (x-N)² Where L (x) = Loss Function, k = C/d² = Constant of proportionality, where C – Loss associated with sp limit d - Deviation of specification from target value x = Quality Features of selected product, N = Nominal Value of the product and (x-N) = Tolerance

Taguchi uses Quadratic Equation to determine loss Curve

In the case of the SONY colour TV sets, let the adjustment cost be A= 300 Rs, when the colour density is out of the tolerance specifications. Therefore, the value of k can be calculated by the following equation: k = 300/5² = 12 Rs

Therefore, the loss function is L(y) = 12(y – m)²This equation is still valid even when only one unit of product is made.

The mean square deviation of objective characteristics from their target values can be applied to estimate the mean value of quality loss Equation. One can calculate the mean square deviation from target σ² (σ² in this equation is not variance) by the following equation (the term σ² is also called the mean square error or the variation of products): σ² =mean value of (y-m)² So L(y) = k(y-m)² becomes L = kσ²

Quality level of Sony TV set

We can evaluate the differences of average quality levels between the TV sets from Japanese Sony and those from San Diego Sony.

it is clear that although the defective ratio of the Japanese Sony is higher than that of the San Diego Sony, the quality level of the former is 3 times higher than the latter.

Japan M 10/6 10²/36 33.33 0.27%

San Diego

M 10/√12 10²/12 100 0.00

Plant

Location

Mean Value of Objective Function

Standard Deviation

Variation Loss L

(in Rs)

Defective Ratio

If Tighten the tolerance

Assume that one can tighten the tolerance specifications of the TV sets of San Diego Sony to be m ± 10/3. Also assume that these TV sets remain uniformly distributed after the tolerance specifications are tightened. The average quality level of San Diego Sony TV sets would be improved to the following quality level:

L = 12[(1/ √12) (10) (2/3)] ² = 45Rs

 Where the value of the loss function is considered the relative quality level of the product. This average quality level of the TV sets of San Diego Sony is 56Rs lower than the original quality level but still 11Rs higher than that of Japanese Sony TV sets. In addition, in this type of quality improvement, one must adjust the products that are between the two tolerance limits,m ± 10/3 and m ± 5, to be within m ± 10/3. In the uniform distribution shown in Figure, 33.3% would need adjustment, which would cost 300Rs per unit. Therefore each TV set from San Diego Sony would cost an additional 100Rs on average for the adjustment: (300)(0.333) = 100RsConsequently, it is not really a good idea to spend 100Rs more to adjust each product, which is worth only 56Rs.

A better way is to apply quality management methods to improve the quality level of products.

Example of loss function

Suppose that the specification on a part is 0.500 ± 0.020 cm. A detailed analysis of product returns and repairs has discovered that many failures occur when the actual dimension is near the extreme of the tolerance range (that is, when the dimensions are approximately 0.48 or 0.52) and costs $50 for repair. Thus, in Equation, the deviation from the target, x – T , is 0.02 and L(x) = $50. Substituting these values, we have:

50 = k(0.02)2 or

k = 50/0.0004 = 125,000

Therefore, the loss function for a single part is L(x) = 125000(x – T)2.

This means when the deviation is 0.10, the firm can still expect an average loss per unit of:

L(0.10) = 125,000(0.10)2 = $12.50 per partKnowing the Taguchi loss function helps designers to determine appropriate tolerances economically. For example, suppose that a simple adjustment can be made at the factory for only $2 to get this dimension very close to the target.

If we set L(x) = $2 and solve for x – T, we get:2 = 125000(x – T)2x – T = 0.004

Therefore, if the dimension is more than 0.004 away from the target, it is more economical to adjust it at the factory and the specifications should be set as 0.500 ± 0.004.

Variation of the Quadratic Loss Function

1) Nominal the best type: Whenever the quality characteristic y has a finite target value, usually nonzero, and the quality loss is symmetric on the either side of the target, such quality characteristic called nominal-the-best type. This is given by equation

  L(y) =k(y-m)²

Example: Color density of a television set and the out put voltage of a power supply circuit.

2)Smaller-the-better type: Some characteristic, such as radiation leakage from a microwave oven, can never take negative values. Also, their ideal value is equal to zero, and as their value increases, the performance becomes progressively worse. Such characteristic are called smaller-the-better type quality characteristics. Examples: The response time of a computer, leakage current in electronic circuits, and pollution from an automobile.In this case m = 0  L(y) =ky²  This is one side loss function because y cannot take negative values.

3)Larger-the-better type: Some characteristics do not take negative values. But, zero is there worst value, and as their value becomes larger, the performance becomes progressively better-that is, the quality loss becomes progressively smaller. ,, also Their ideal value is infinity and at that point the loss is zero. Such characteristics are called larger-the-better type characteristics.Example: Such as the bond strength of adhesives.Thus we approximate the loss function for a larger-the-better type characteristic by substituting 1/y for y in L(y) = k [1/y²]

4)Asymmetric loss function: In certain situations, deviation of the quality characteristic in one direction is much more harmful than in the other direction. In such cases, one can use a different coefficient k for the two directions. Thus, the quality loss would be approximated by the following asymmetric loss function:   k(y-m) ²,y>m L(y) = k(y-m) ², y≤m

Introduction to Analysis of variation(ANOVA)

What is ANOVA  ANOVA is a statistically based decision tool for detecting

any differences in average performance of groups of items tested.

 ANOVA is a mathematical technique which breaks total

variation down into accountable sources; total variation is decomposed into its appropriate components.

Degrees of Freedom (dof)

Degree of freedom are the number of observations that can be varied independently of each other.

Two –Way ANOVA

There are two controlled parameters in this experimental situation  Let us consider an experimental situation. A student worked at an aluminum casting foundry which manufactured pistons for reciprocating engines.

 The problem with the process was how to attain the proper hardness (Rockwell B) of the casting for a particular product. Engineers were interested in the effect of copper and magnesium content on casting hardness.According to specifications the copper content could be 3.5 to 4.5% and the magnesium content could be 1.2 to 1.8%.

Two –Way ANOVA continue…

The student runs an experiment to evaluate these factors and conditions simultaneously.  If A = % Copper Content = 3.5 = 4.5If B = % Magnesium Content = 1.2 = 1.8 The experimental conditions for a two level factors is given by = 4 which are Imagine, four different mixes of metal constituents are prepared, casting poured and hardness measured. Two samples are measured from each mix for hardness. The result will look like:

76, 78 73, 74

77, 78 79, 80

2A1B 2B1A

1A 2A

1B2B

Two –Way ANOVA continue…

To simplify discussion 70 points from each value are subtracted in the above observations from each of the four mixes. Transformed results can be shown as:

6, 8 3, 4

7, 8 9, 10

1A 2A

1B

2B

Two way ANOVA calculations: 1) The variation may be decomposed into more

components:2) Variation due to factor A3) Variation due to factor B4) Variation due to interaction of factors A and B5) Variation due to error

Two –Way ANOVA continue…

An equation for total variation can be written asA x B represents the interaction of factor A and B. The interaction is the mutual effect of Cu and Mg in affecting hardness. Some preliminary calculations will speed up ANOVA

Total6, 8 3, 4 217, 8 9, 10 34

Total 29 26 55

1A 2A

1B2B

Grand Total

41 An 41 Bn 42 An 42 Bn

=29 =26 =21 =34

T = 55, N = 8

1A 2A 1B 2B

Two –Way ANOVA continue…

Total Variation

N

TySS

N

iiT

22

1

)(

8

552

= 6² + 8² + 3² + ----------- + 10² - = 40.85

N

T

n

ASS

Ak

i Ai

iA

2

1

2

N

T

n

A

n

A

n

A

Ak

k

AA

22

2

22

1

21

Variation due to factor A

ASS8

55

4

26

4

29 222

= 1.125=

Mathematical check : Numerator 29 + 26 = 55 and Denominator 4 + 4 = 8

Two –Way ANOVA continue…

For a two level experiment, when the sample sizes are equal, the equation above can be simplified to this special formula:

N

AASSA

221

8

)2629( 2 = = 1.125

Similarly the variation due to factor B

125.21

8

)3421( 2221

N

BBSSB

To calculate the variation due to interaction of factors A and BLet represent the sum of data under the ith condition of the combination of factor A and B. Also let c represent the number of possible combinations of the interacting factors and the number of data under this condition.

iBA )(

iBAn )(

Two –Way ANOVA continue…

BA

c

i iBA

iBA SSSS

N

T

n

BASS

2

1 )(

2)(

Note that when the various combinations are summed, squared, and divided by the number of data points for that combination, the subsequent value also includes the factor main effects which must be subtracted. While using this formula, all lower order interactions and factor effects are to be subtracted. For the example problem: = 14, = 15, = 7 = 19

And no. of possible combinations c = 4And since there are two observations under each combination

= 2

11BA 21BA 12BA 22BA

iBAn )(

Two –Way ANOVA continue…

BABA SSSSSS 8

55

2

19

2

15

2

7

2

14 22222

= 15.125

eBABAT SSSSSSSSSS Since

500.3125.15125.21125.1875.40 eSS

Degrees of Freedom (Dof) – Two way ANOVA

eBABAt vvvvv

= N – 1 = 8 – 1 = 7tvAv Ak - 1 = 1=

Bv Bk - 1 = 1=

1))(( BABA vvv

41117 ev

Two –Way ANOVA continue…

ANOVA summary Table (Two-way)

Source SS Dof v Variance V FA 1.125 1 1.125 1.29B 21.125 1 21.125 24.14*A x B 15.125 1 15.125 17.29**E 3.500 4 0.875Total 40.875 7

* at 95% confidence** at 90% confidence

The ANOVA results indicate that Cu by itself has no effect on the resultant hardness, magnesium has a large effect (largest SS) on hardness and the interaction of Cu and Mg plays a substantial part in determining hardness.

Two –Way ANOVA continue…

Geometrically, there is some information available from the graph that may be useful. The relative magnitudes of the various effects can be seen graphically. The B effect is the largest, A x B effect next largest and the A effect is very small.  See

1.74

291 A 5.6

4

262 A

2.54

211 B 5.8

4

342 B

Two –Way ANOVA continue…

Hard-ness

2

4

6

8

10

B1

B2

B effect

A x B effect

A effect

A2A1

Mid pt. A2B1 & A2B2

Mid pt. on line B1

Introduction To Orthogonal Array

Engineers and Scientists are often faced with two product or process improvement situations. 1st development situation is to find a parameter that

will improve some performance characteristic to an acceptable and optimum value. This is the most typical situation in most organizations.

2nd situation is to find a less expensive, alternate design, material, or method which will provide equivalent performance

Orthogonal Array

WHAT IS ARRAYAn Array’s name indicates the number of rows and column it has , and also the number of levels in each of the column. Thus the array L4 has four rows and three 2-level column.WHAT IS ORTHOGONALITYOne main requirement of orthogonality is a balanced experiment which means equal number of samples under various test conditions.

History of Orthogonal Array

F1 F2 F3 F4

I1

I2

I3

I4

F1 F2 F3

F4

I1

A1 A2

A3

A4

I2 A2 A3

A4

A1

I3

A3 A4

A1

A2

I4

A4 A1

A2

A3

Most common test plan is to evaluate the effect of one parameter on product performance. This is what is typically called as one factor experiment.

This experiment evaluates the effect of one parameter while holding everything else constant. The simplest case of testing the effect of one parameter on performance would be to run a test at two different conditions of that parameter.

Trial No Factor Level Test Results1 1(LOW) *

2 2(HIGH) *

Several Factors One at a Time(OFAT)

Speed Feed Depth Material Test Results

Trial No. A B C D

1 1 1 1 1 *

2 2 1 1 1 *

3 1 2 1 1 *

4 1 1 2 1 *

5 1 1 1 2 *

1-LOW LEVEL2-HIGH LEVEL

OFAT

Isolate what is believed to be the most important factor

Investigate this factor by itself, ignoring all others

Make recommendations on changes to this crucial factor

Move on to the next factor and repeat

ExampleA process producing steel springs is generating considerable scrap

due to cracking after heat treatment. A study is planned to determine better operating conditions to reduce the cracking problem.

 There are several ways to measure crackingSize of the crackPresence or absence of cracks The response selected was Y: the percentage without cracks in a batch of 100 springs Three major factors were believed to affect the responseT: Steel temperature before quenchingC: carbon content (percent)O: Oil quenching temperature

Investigating Many factor

Factor Low (Level1) High(Level2)T 1450 degree 1600 degree

C 0.5% 0.7%

O 70 degree 120 degree

Traditional Approach

Steel Tempt

%age of spring without crack Average

1450 61 67 66 68 65.5

1600 79 75 71 77 75.5

 

Carrying out similar OFAT experiments for C and O would require a total of 24 observations and provide limited knowledge.Conclusion: Increased T reduces cracks by 10% 

Four runs at each level of T with C and O at their low levels

Factorial Approach

Run C T O1 0.5 1450 70

2 0.7 1450 70

3 0.5 1600 70

4 0.7 1600 70

5 0.5 1450 120

6 0.7 1450 120

7 0.5 1600 120

8 0.7 1600 120

Include all factors in a balanced design: To increase the generality of the conclusions,

use a design that involves all eight combinations of the three factors.

The above eight runs constitute a FULL FACTORIAL DESIGN. The design is balanced for every factor. This means 4 runs have T at 1450 and 4 have T at 1600. Same is true for C and O.

Advantages Of Factorial approach over OFAT

The effect of each factor can be assessed by comparing the responses from the appropriate sets of four runs.More general conclusions8 runs rather than 24 runs

Efficient Test Strategy

Factor Level 1 Level 2

Front cover design Production New

Gasket Design Production New

Front bolt torque Low High

Gasket Coating Yes No

Pump Housing Finish Rough Smooth

Rear bolt torque Low High

Torque pattern Front rear Rear front

A Full Factorial Experiment

A1 A2B1 B2 B1 B2C1 C2 C1 C2 C1 C2 C1 C2D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 D1 D2

E1F1

G1G2

F2G1G2

E2F1

G1G2

F2G1G2

If a full factorial is to be used in this situation will have to be conducted. (As shown in figure below)

Trial No.

Column No.

1 2 3 4 5 6 7

1 1 1 1 1 1 1 12 1 1 1 2 2 2 23 1 2 2 1 1 2 24 1 2 2 2 2 1 15 2 1 2 1 2 1 26 2 1 2 2 1 2 17 2 2 1 1 2 2 18 2 2 1 2 1 1 2

This is 1/16th FFE which has only 8 of the possible 128 combinations represented

Example Of Orthogonal Array

During the late 1980s, Modi Xerox had a large base of customers (50 thousand) for this model spread over the entire country. Many buyers of these machines earn their livelihood by running copying services. Each of these buyers ultimately serves a very large number of customers (end user). When copy quality is either poor or inconsistent, customers earn a bad name and their image and business gets affected.

A Case Study Of Orthogonal Array

The pattern of blurred images (skips) observed in the copy is shown in Figure above. It usually occurs between 10-60 mm from the lead edge of the paper. Sometimes, on a photocopy taken on a company letterhead paper, the company logo gets blurred, which is not appreciated by the customer. This problem was noticed in only one-third of the machines produced by the company, with the remaining two-thirds of machines in the field working well without this problem. The in-house test evaluation record also confirmed the problem in only about 15% of the machines produced. Data analysis indicated that not all the machines produced were faulty. Therefore, the focus of further investigation was to find out what went wrong in the faulty machines or whether there are basic differences between the components used in good and faulty machines.

Selection of Factor and Intersection To EvaluateA copier machine consists of more than 1000 components and assemblies. A brainstorming session by the team helped in the identification of 16 components suspected to be responsible for the problem of blurred images. Each Suspected component had at least two possible dimensional characters which could have resulted in the skip symptom. This led to more than 40 probable causes (40 dimensions arising out of 16 components) for the problem. An attempt was made by the team to identify the real causes among these 40 probable causes. Ten bad machines were stripped open and various dimensions of these 16 component were measured. It was observed that all the dimensions were well within specifications Hence, this investigation did not give any clues to the problem. Moreover, the time and effort spent in dismantling the faulty machines and checking various dimensions in 16 components was in vain. This gave rise to the thought that conforming to specification does nor always lead to perfect quality. The team needed to think beyond the specification in order to find a solution to the problem

Selection of No. of factorAn earlier brainstorming session had identified 16 components that were likely to be the cause of this problem. A study of travel documents of 300 problem machines revealed that on 88% of occasions, the problem was solved by replacing one or more of only four parts of the machine. These four parts were from the list of 16 parts identified earlier. They were considered to be Critical and it was decided to conduct an experiment on these four parts. These parts were the following:(a) Drum shaft (b) Drum gear (c) Drum flange (d) Feed shaftTwo sets of these parts Were taken for Experiment I, one from an identified problem machine and one from a problem free machine. The two levels considered in the experiment were good and bad; ‘good’ signifying parts from the problem-free machine and ‘bad’ signifying parts from the problem machine. The factors and levels thus identified are given in Table below.

Selection of no. of factor and level

Selection of appropriate Orthogonal Array

Assignment of Factor/Interaction to Columns

Analysis and Results

Online quality control Off- line quality control

Types of Quality Control

ROBUST DESIGNING

What is Robustness

Robustness means small variation in performance.

All products look good when they are precisely made.

Robust products work well . Only robust products provide consistent

customer satisfaction .

Example of Robustness

Suppose two shooter “X” and “Y” go to the target range, and each shoots an initial round of 10 shots. X has his shots in a tight cluster, which lies outside the bull’s-eye, but Y actually has one shot in the bull’s-eye, but his success results only from his hit-or-miss pattern.

“X” “Y” In this initial round Y has one By more bull’s-eye than X , but X is the robust

shooter. a simple adjustment of his sights, X will move his tight cluster into the bull’s-eye for the next round. Y faces a much more difficult task. He must improve his control altogether, systematically optimizing his arm position, the tension of his spring, and other critical parameters.

Why we use Robust Design

Every customer wants a product with high quality and a minimum cost, this is achieved by robust designing. There are mainly four types of cost which is under the category of obvious quality costs.

• Internal quality cost• External quality cost• Appraisal quality cost• Prevention cost

These obvious quality cost are incurred directly by the producer and then passed on to the customer through the purchase price of the product. Robust Design is a symmetric method for keeping the producer’s cost low while delivering a high quality product. This is done by ROBUSTNESS STRATEGY.

Tools for robustness strategy

Signal to Noise Ratio P- Diagram Quadratic loss function Orthogonal Array

Signal to Noise Ratio (S/N)

Signal to noise ratio used for predicting the field quality.

S/N= amt. of energy for intended function amt. of energy wasted

Example of Signal to Noise ratio When a person puts his foot on the brake pedal

of the car, energy is transformed with the intent to slow car, which is the signal.

However some of the energy is wasted by pad wear, squeal, heat etc. these are called noise.

Energy transformation

Slow car

wear

squeal

heat

Etc.

Signal

Noise

Brake

Signal factors are set by the designer/ operator to obtain the intended value of the response/ output variable.

Noise factors are not controlled by the designer/ operator or very difficult and expensive to control.

Purpose of noise factors: To make the product /process robust against Noise Factors(NF)

Types of Noise Factors

Internal Noise External Noise

Internal Noise: These are mainly due to deterioration such as product wear, very old material, changes in components or material with time or use.External Noise: These are due to variation in environmental conditions such as dust, temperature, humidity etc.

Types of Signal to Noise ratio(S/N)

Smaller the Better Larger the Better Nominal the Best

Signal to Noise Ratio

Smaller-the – Better(S/Ns) The S/Ns ratio for Smaller the Better is used where the smaller

value is desired. In this the target value is zero.

Larger-the –Better(S/NL) The S/NL ratio for Larger the Better is used where the largest

value is desired. In this the target value is also zero.

Nominal –the- Best(S/NN) The S/NN ratio for Nominal the better is used where the

Nominal or Target value and variation about that value is minimum.

Here target value is finite not zero.

P- Diagram P- Diagram is a permanent diagram for

product/ process. It is must for every development project.

P-Diagram defines clearly and deeply the scope of development.

• In P-Diagram, first we identify the signal (Input) and response (output) associated with design concept for Robust Design.

For example: In designing the cooling system for a room the thermostat setting is the signal and the resulting room temperature is the response.

Next consider the parameters/factors that are beyond the control of the designer. Those factors are called noise factors. Outside temperature, opening/closing of windows, and number of occupants are examples of noise factors. Parameters that can be specified by the designer are called control factors. The numbers of registers, their locations, size of the air conditioning unit, insulation are examples of control factors.

Ideally, the resulting room temperature should be equal to the set point temperature. Thus the ideal function here is a straight line of slope one in the signal-response graph. This relationship must hold for all operating conditions. However, the noise factors cause the relationship to deviate from the ideal.

The job of the designer is to select appropriate control factors and their settings so that the deviation from the ideal is minimum at a low cost. Such a design is called a minimum sensitivity design or a Robust Design.